On Apery constants of homogeneous varieties. 1. Introduction The

On Apery constants of homogeneous varieties. S. GALKIN Abstract. We do numerical computations of Apery constants for homogeneous varieties G/P for maximal parabolic groups P in Lie groups of type An , n 6 10, Bn , Cn , Dn , n 6 7, E6 , E7 , E8 , F4 and G2 . These numbers are identified to be polynomials in the values of Riemann zeta-function ζ(k) for natural arguments k > 2.

1. Introduction The article is devoted to the computations of Apery numbers for the quantum differential equation of homogeneous varieties, so first we introduce these 3 notions. Let X be a Fano variety of index r: −KX = rH, and q be a coordinate on the anticanonical d be an invariant vector field. Cohomologies torus Z − KX ⊗ C∗ = Gm ∈ Pic(X) ⊗ C∗ , and D = q dq q H (X) are endowed with the structure of quantum multiplication ?, and associativity of ? implies that first Dubrovin’s connection given by (1.1)

Dφ = H ? φ

is flat. If we replace in equation 1.1 quantum multiplication with the ordinary cup-product, then it’s q solutions are constant Lefschetz coprimitive (with respect to H) classes in H (X). Dimension µ of the space of homolorphic solutions of 1.1 is the same and equal to the number of admissible initial conditions (of the recursion on coefficients) modulo q, i.e. the rank of the kernel of cupq multiplication by H in H (X), that is the dimension of coprimitive Lefschetz cohomologies. Solving equation 1.1 by Newton’s method one obtains a matrix-valued few-step recursion reconstructing all the holomorphic solutions from these initialPconditions. Givental’s theorem states that the solution A = 1 + n>1 a(n) q n associated with the primitive class 1 ∈ H 0 (X) is the I-series of the variety X (the generating function counting some rational curves of X). Choose a basis of other solutions A1 , . . . , Aµ−1 associated with homogeneous primitive classes of nondecreasing codimension 1 P P (n) Put A = n>0 a(n) tn and Ai = n>0 ai tn . We call the number (n)

ai lim (n) n→∞ a i-th Apery constant after the renown work [2], where ζ(3) and ζ(2) were shown to be of that kind for some differential equations and such a presentation was used for proving the irrationality of these two numbers. If there is no choosen basis, for any coprimitive class γ one still may consider 1One

could also consider other bases, e.g. it is often exists a base with ith element Bi determined by the condition Bi = ti (modtµ ). But the answer in this base looks worse. Finally one may reject to choose any basis and express everything invariantly in the dual space of primitive classes. 1

the solution Aγ =

P

n>1

(n)

aγ q n = P r0 (γ +

P

n>1

(n)

Aγ q n ) and the limit (n)

aγ Apery(γ) = lim (n) n→∞ a

(1.2)

Defined in that way, Apery is a linear map from coprimitive cohomologies to C. A linear map on coprimitive cohomologies is dual 2 to some (nonhomogeneous) primitive cohomology class with coefficients in C. We name it Apery characteristic class A(X) ∈ H 6dim X (X, C). Consider the homogeneous ring R = Q[c1 , c2 , c3 , . . . ], deg ci = i and a map ev : R → C sending c1 to Euler constant C 3, and ci to ζ(i). The main conjecture we verify is the following q

Conjecture 1.3. Let X be any Fano variety and γ ∈ H (X) be some coprimitive with respect to −KX homogeneous cohomology class of codimension n. Consider two solutions of quantum D-module: A0 associated with 1 and Aγ associated with γ. Then Apery number for Aγ (i.e. (k)

limk→∞

aγ

(k)

a0

) is equal to ev(fγ ) for some homogeneous polynomial fγ ∈ R(n) of degree n.

Actually, in our case there is no Euler constant contributions, and the conjecture seems too strong to be true - it would imply that some of differential equations studied in [1] has nongeometric origin (at least come not from quantum cohomology), because their Apery numbers √ does not seem to be of the kind described in the conjecture (e.g. Catalan’s constant, π 3 , π 3 3). From the other point of view, for toric varieties X the solutions of QDE are known to be pullbacks of hypergeometric functions, coefficients of hypergeometric functions are rational functions of Γvalues, and the Taylor expansion (1.4)

log Γ(1 + x) = Cx +

X ζ(k) k>2

k

xk

suggests all Apery constants would probably be rational functions in C and ζ(k). So whether one believes in toric degenerations or hypergeometric pullback conjecture, he would find natural to √ 1 91 3 believe in 1.3. Also Apery limits like 432 ζ(3) − 216 π 3 may appear as ”square roots” or factors 3 912 2 6 (convolutions with quadratic character or something) of geometric ones like 432 2 ζ(3) − 2162 π . This is not even the second paper (the computations of this paper were described by Golyshev 2-3 years ago) discussing the natural appearance of ζ-values in monodromies of QDEs. In case of fourfolds X the expression of monodromies in terms of ζ(3), ζ(2k) and characteristic numbers of anticanonical section of X was given by van Straten [14], Γ-class for toric varieties appears in Iritani’s work [9], and in general context in [10]. Let G be a (semi)simple Lie group, W be it’s Weyl group, P be a (maximal) parabolic subgroup associated with the subset (or just one) of the simple roots of Dynkin diagram, and denote factor G/P by X. X is a homogeneous Fano variety with rk Pic X equal to the number of chosen roots. In case when G is simple and P is maximal we have Pic X = ZH, where H is an ample generator, KX = −rH. For homogeneous varieties with small number of roots in Dynkin diagram (being more precise, with not too big total dimension of cohomologies) by the virtue of Peterson’s version of Quantum 2One 3C

may choose between Poincare and Lefschetz dualities. We prefer the first one. Pn = limn→∞ ( k=1 k1 ) − ln n 2

Chevalley formula [4][Theorem 10.1] we explicitly compute the operator H? 4, and hence find 1.1 (k)

with all it’s holomorphic solutions. Then we do a numerical computation of the ratios

aγ

(k)

a0

for big

k (e.g. k = 20 or 40 or 100), and guess the values of the corresponding Apery constants, then state some conjectures (refining 1.3) on what these numbers should be. 2. Grassmanian Gr(2,N) Let V be the tautological bundle on Grassmanian Gr(2, N ), consider H = c1 (V ) and c2 = c2 (V ). q Cohomologies H (Gr(2, N ), C) is a ring generated by H and c2 with relations of degree > N − 1. So there is at least 1 primitive (with respect to H) Lefschetz cohomology class p2k in every even codimension 2k, 0 6 k 6 N 2−2 . Since N −2   X 2 N (2N − 3 − 4k) dim H (Gr(2, N ), C) = = 2 k=0

q

they exhaust all the primitive classes. p0 = 1 c2 · c12N −6 2 p2 = c2 − c1 c12N −4 ... The associated conjectural Apery numbers are listed in the following table, Apery numbers associated with the primitive cohomology classes of codimension 2k are rational multiples of ζ(2k) 'Q∗ π 2k . X Gr(2, 4) Gr(2, 5) Gr(2, 6) Gr(2, 7) Gr(2, 8) Gr(2, 9) Gr(2, 10) Gr(2, 11) Gr(2, 12) Gr(2, 13) Gr(2, 14) Gr(2, 15)

µ p2 p4 2 0 2 ζ(2) 3 2ζ(2) 0 27 ζ(4) 3 3ζ(2) 4 4 4ζ(2) 16ζ(4) 4 5ζ(2) 111 ζ(4) 4 5 6ζ(2) 42ζ(4) 5 7ζ(2) 235 ζ(4) 4 6 8ζ(2) 78ζ(4) 6 9ζ(2) 399 ζ(4) 4 7 10ζ(2) 124ζ(4) 7 11ζ(2) 603 ζ(4) 4

p6

p8

0 675 ζ(6) 16

108ζ(6) 3229 ζ(6) 16 328ζ(6) 7855 ζ(6) 16 695ζ(6) 15113 ζ(6) 16

0 18375 ζ(8) 64

768ζ(8), 96111 ζ(8), 64 7664 ζ(8), 3 768085 ζ(8), 192

Remark 2.1. Gr(2, 5) case is essentially Apery’s recursion for ζ(2) (see remark 7.1). 4We

used computer algebra software LiE [11] for the computations in Weyl groups. The script is avaiable at http://www.mi.ras.ru/∼galkin/work/qch.lie, and the answer is available in [6]. We used PARI/GP computer algebra software [12] for solving the recursion and finding the linear dependencies between the answers and zetapolynomials. Script for this routine is available at http://www.mi.ras.ru/∼galkin/work/apery.gp. 3

Remark 2.2. Constants for p2 depend linearly on N , constants for p4 depend quadratically on N , constants for p6 looks like they grow cubically in N . So we conjecture constants for p2k is ζ(2k) times polynomial of degree k of N . The proof for the computation of p2 (in slightly another Q-basis) was given recently in [7]. Let us describe a transparent generalization of this method for the all primitive p2k of Gr(2, N ). Quantum D-module for Gr(r, N ) is the r’th wedge power of quantum D-module for PN −1 (solutions of QDE for Gr(r, N ) are r × r wronskians of the fundamental matrix of solutions for PN −1 ). Let N be either 2n or 2n + 1. Consider the deformation of quantum differential equation for PN −1 : (2.3)

(D − u1 )(D + u1 )(D − u2 )(D + u2 ) · · · · · (D − un )(D + un ) · DN −2n − q

This equation has (at least) 2n formal solutions: X 1 qk Ra = N −2n Γ(k − u )Γ(k + u ) · · · · · Γ(k − u )Γ(k + u ) · Γ(k) 1 1 n n k−a∈Z +

0 for a = u1 , −u1 , . . . , un , −un . Let Si = Ru0 i R−ui − R−u Rui be the wronskians. Then Si = i P (k) k k>0 si q for i = 1, . . . , n are n holomorphic solutions of the wedge square of the deformed equation 2.3. Using his explicit calculation for the monodromy of hypergeometric equation 2.3 and Dubrovin’s theory, Golyshev computes the monodromy of ∧2 (2.3) and demonstrates the formula of sinuses: (k)

(2.4)

lim

si

k→∞ s(k) j

=

sin(2πui ) sin(2πuj )

sin(2πui ) So in the base of S1 , . . . , Sn Apery numbers are sin(2πu . One then reconstructs the required 1) Apery numbers by applying the inverse fundamental solutions matrix to this vector of sinuses, and limiting all ui to 0.

3. Other grassmannians of type A Let V be the tautological bundle on Grassmanian Gr(3, N ), consider H = c1 (V ), c2 = c2 (V ) and c3 = c3 (V ). q Cohomologies H (Gr(3, N ), C) are generated by H, c2 and c3 with relations of degree > N − 2. q In particular, if N > 7, then 1, c2 , c3 , c22 and c2 c3 generate H 610 (X, Q) = H (X)/H >10 (X) as Q[c1 ]-module. So there is 1 primitive class in codimensions 0,2,3,4 and 5. X µ p2 p3 p4 p5 p>6 Gr(3, 6) 3 0 −6ζ(3) Gr(3, 7) 4 ζ(2) −7ζ(3) − 17 ζ(4) − 49 ζ(3)2 − 945 ζ(6) 4 2 16 Gr(3, 8) 5 2ζ(2) −8ζ(3) 0 −8ζ(2)ζ(3) − 4ζ(5) −32ζ(3)2 − 62ζ(6) 27 ±( 81 Gr(3, 9) 8 3ζ(2) −9ζ(3) ζ(4) − 27 ζ(2)ζ(3) − 92 ζ(5) ζ(3)2 + 871 ζ(6)), . . . 4 2 2 16 Gr(3, 10) 10 4ζ(2) −10ζ(3) 16ζ(4) −20ζ(2)ζ(3) − 5ζ(5) ±(50ζ(3)2 + 32ζ(6)), . . . Gr(3, 11) 13 5ζ(2) −11ζ(3) 111 ζ(4) − 55 ζ(2)ζ(3) − 11 ζ(5) (− 121 ζ(3)2 + 110 ζ(6)) ± 45 ζ(6), . . . 4 2 2 2 16 16 Remark 3.1. One may notice that the Apery constants of p2 and p4 for Gr(3, N ) are equal to the Apery constants of p2 , p4 for Gr(2, N − 2). Why? Is it possible to make an analogous statement for p6 (obviously one should choose another basis of two elements in H 12 (Gr(3, N )) to vanish appearing ζ(3)2 terms)? 4

Remark 3.2. p2 is linear of N , p4 is quadratic of N , p3 is linear of N , p5 is quadratic of N . Remark 3.3. p5 is quadratic polynomial of N times ζ(2)ζ(3) plus linear polynomial of N times ζ(5) . This gives a suggestion on a method of separating e.g. ζ(4) and ζ(5). Actually it is − p2 p3 −N 2 ζ(2)2 in p4 — ζ(4) term should be only linear and ζ(2)2 is quadratic in N . Similarly the coefficient p2 at ζ(3)2 is quadratic in N (and in the choosen basis p6 ’th ζ(3)2 -part is 23 ). For Gr(4, N ) we still do have a unique primitive class of codimension 5. X Gr(4, 8)

µ 8

p2 0

p3 −8ζ(3)

p4 −6ζ(4)

p04 0

p5 none

p>6 32ζ(3)2 + 50ζ(6) twice and 08 9 81 Gr(4, 9) 12 ζ(2) −9ζ(3) 21 ζ(4) − ( ζ(4) (ζ(2)ζ(3) + ζ(5)) ζ(3)2 + 117 ζ(6))± 159 ζ(6), 4 2 2 4 16 ... Gr(4, 10) 18 2ζ(2) −10ζ(3) −2ζ(4) 2ζ(4) −10ζ(2)ζ(3) − 5ζ(5) 50ζ(3)2 + 31ζ(6), 50ζ(3)2 , 06 , . . . 33 11 121 Gr(4, 11) 24 3ζ(2) −11ζ(3) 15 ζ(4) 3ζ(4) − ζ(2)ζ(3) − ζ(5) ( ζ(3)2 + 35 ζ(6))± 197 ζ(6), 4 2 2 2 2 16 27 ζ(6),. . . 16 Remark 3.4. Apery of p3 for Gr(3, N ) and Gr(4, N ) coincide. Apery of p2 for Gr(4, N ) is equal to Apery of p2 for Gr(3, N − 2) and Apery of p2 for Gr(2, N − 4). For Gr(5, 10) we have 20 Lefschetz blocks, they correspond to 20 solutions, and hence 19 Apery constants. Some of them vanish, while some other coincide (because solutions differ only by some character). X µ p2 p3 p4 p04 p5 p05 Gr(5, 10) 20 0 −10ζ(3) −6ζ(4) 0 10ζ(5) −10ζ(5) Gr(5, 11) 32 ζ(2) −11ζ(3) − 21 ζ(4) 11(ζ(5) − ζ(2)ζ(3)) −11ζ(5) ζ(4) 4 4. B,C,D cases The picture for other 3 series of classical groups is similar. For 1 6 k 6 n let D(n, k) denote homogeneous space of isotropic (with respect to nondegenerate quadratic form) k-dimensional linear spaces in 2n-dimensional vector space. D(n, k) = OGr(k, 2n) = G/P where G is Spin(2n), and maximal parabolic subgroup P ⊂ G corresponds to k’th simple root counting from left to right. Similarly define B(n, k) = OGr(k, 2n + 1) and C(n, k) = SGr(k, 2n). X B(3, 2) B(4, 2) B(4, 3) B(4, 4) B(5, 2) B(5, 3)

µ 2 3 3 2 4 8

B(5, 4)

8

Apery numbers −2ζ(2). ζ(2), − 41 ζ(4). 2 −4ζ(2), −4ζ(3). 2ζ(3). 3ζ(2), 23 ζ(4) − 1191 ζ(6). 8 02 , −8ζ(3), −24ζ(4), 20ζ(5), 64 ζ(3)2 + 80 ζ(6), 32ζ(3)ζ(4) + 232 ζ(7), 256 ζ(3)3 + 3 3 3 21 320 ζ(3)ζ(6) − 480 ζ(4)ζ(5) − 1000 ζ(9). 7 7 21 −6ζ(2), −6ζ(3), −45ζ(4), 9ζ(2)ζ(3) + 21ζ(5), 15ζ(3)2 + 1141 ζ(6), 56ζ(2)ζ(5) + 24 171 222 263 3 30ζ(3)ζ(4) + 52ζ(7), 266 ζ(3) − ζ(2)ζ(7) − ζ(3)ζ(6) − ζ(4)ζ(5) + 136 ζ(9). 5 5 5 5 5 5

X µ B(5, 5) 3 B(6, 2) 5 B(6, 3) 12

Apery numbers 4ζ(3), 20ζ(5). 5ζ(2), 87 ζ(4), − 485 ζ(6), − 35073 ζ(8). 4 8 32 2ζ(2), −6ζ(3), −12ζ(4), −12ζ(2)ζ(3) + 18ζ(5), −36ζ(3)2 − 146ζ(6), 36ζ(3)2 + 2 , 803ζ(3)3 − 2ζ(6), 24ζ(2)ζ(5) + 24ζ(3)ζ(4) + 76ζ(7), 360ζ(3) ζ(2)−1080ζ(3)ζ(5)+1176ζ(8) 11 528ζ(2)ζ(7) + 318ζ(3)ζ(6) − 244ζ(4)ζ(5) − 35ζ(9), 75ζ(3)3 − 336ζ(2)ζ(7) − 395ζ(3)ζ(6) − 22ζ(4)ζ(5) − 70ζ(9),. . . B(6, 4) 18 −1ζ(2), −10ζ(3), − 17 ζ(4), −14ζ(4), 5ζ(2)ζ(3) + 19ζ(5), 50ζ(3)2 + 317ζ(6), 4 −50ζ(3)2 − 4135 ζ(6), 8 B(6, 5) 14 −8ζ(2), −8ζ(3), −84ζ(4), 64ζ(2)ζ(3) + 16ζ(5), −64ζ(2)ζ(3), 80 ζ(3)2 + 24ζ(6), 3 ζ(3)ζ(4) + 101 ζ(7), 110ζ(2)ζ(5) + 49 2 2 B(6, 6) B(7, 2) B(7, 7)

5 6 8

6ζ(3), 18ζ(5), −18ζ(3)2 − 60ζ(6), 36ζ(3)3 + 360ζ(3)ζ(6) + 332ζ(9) 7ζ(2), 211 ζ(4), 1733 ζ(6), − 76699 ζ(8), − 5368203 ζ(10). 4 8 96 640 ζ(3)3 + 480ζ(3)ζ(6) + 8ζ(3), 16ζ(5), −30ζ(3)2 − 60ζ(6), −112ζ(7), 256 3

992 ζ(9), 3

...

Remark 4.1. B(4, 4) case is essentially Apery’s recursion for ζ(3). X C(3, 2) C(3, 3) C(4, 2) C(4, 3) C(4, 4) C(5, 2) C(5, 3)

µ 2 2 3 4 2 4 8

Apery numbers 2ζ(2). 7 ζ(3). 2 4ζ(2), 16ζ(4). ζ(2), −9ζ(3), − 29 (ζ(2)ζ(3) + ζ(5)). 4ζ(3). 6ζ(2), 42ζ(4), 108ζ(6). 3ζ(2), −11ζ(3), 27 ζ(4), − 33 ζ(2)ζ(3) − 11 ζ(5), 242 ζ(3)2 + 2383 ζ(6), −11ζ(2)ζ(5) − 4 2 2 3 48 11 309 41 99 3 ζ(3)ζ(4) − 3 ζ(7), 108ζ(3) − 38ζ(2)ζ(7) + 4 ζ(3)ζ(6) − 4 ζ(4)ζ(5) + 36ζ(9) 4 C(5, 4) 8 02 , −10ζ(3), 30ζ(4), −5ζ(5), 250 ζ(3)2 + 175 ζ(6), − 100 ζ(3)ζ(4) − 10 ζ(7), 2500 ζ(3)3 + 3 3 3 9 21 ζ(4)ζ(5) − 10 ζ(9). 250ζ(3)ζ(6) − 150 7 21 9 21 C(5, 5) 3 2 ζ(3) , − 2 ζ(5). C(6, 2) 5 8ζ(2), 78ζ(4), 328ζ(6), 768ζ(8). C(6, 3) 12 5ζ(2), −13ζ(3), 111 ζ(4), − 65 ζ(2)ζ(3)− 13 ζ(5), − 169 ζ(3)2 + 155 ζ(6), 169 ζ(3)2 + 65 ζ(6), 4 2 2 2 16 2 2 15 500 131 2 3 C(6, 6) 4 ζ(3), −11ζ(5), −25ζ(3) − 2 ζ(6), 3 ζ(3) + 150ζ(3)ζ(6) − 3 ζ(9). C(7, 2) 6 10ζ(2), 124ζ(4), 695ζ(6), 7664 ζ(8), 5760ζ(10). 3 23 121 15 2 ζ(3), − ζ(5), − ζ(3) − ζ(6), 71 ζ(7), 1331 ζ(3)3 + 165ζ(3)ζ(6) − 263 ζ(9), C(7, 7) 8 11 2 2 4 2 2 6 6 781 529 63 2 ζ(3)ζ(7) − 12 ζ(5) − 2 ζ(10),. . . 12

Remark 4.2. One may notice that Apery numbers for C(2, n) = SGr(2, 2n) coincide with Apery numbers of Gr(2, 2n) except the last 0. The reason for this coincidence is that SGr(2, 2n) is a quadratic hyperplane section of Gr(2, 2n), so by quantum Lefschetz (7.1) has almost the same Apery numbers. Remark 4.3. For general k spaces OGr(k, N ) and SGr(k, N ) are sections of ample vector bundles over Gr(k, N ) (symmetric and wedge square of tautological bundle). Is it possible to formulate a generalization of quantum Lefshetz principle explaining the relations between Apery numbers of OGr(k, N ), SGr(k, N ) and Gr(k, N )? 6

X µ D(4, 2) 4 D(5, 2) 5 D(5, 3) 9 D(5, 4) 2 D(6, 2) 6 D(6, 3) 14 D(6, 5) D(7, 2) D(7, 6)

3 7 5

Apery numbers 0, 0, −24ζ(4). 2ζ(2), 0, −12ζ(4), −144ζ(6). ζ(4), 3ζ(2)ζ(3) + 21ζ(5), 05 , 12ζ(3)2 + 275 ζ(6). −ζ(2), −ζ(2), −6ζ(3), 04 , − 45 2 24 2ζ(3). 4ζ(2), 10ζ(4), 10ζ(4), −124ζ(6), −960ζ(8). ζ(2), −5ζ(3), −5ζ(3), − 41 ζ(4), 0, −5ζ(2)ζ(3) + 19ζ(5), 25 ζ(3)2 + 953 ζ(6), 25 ζ(3)2 − 2 2 16 2 937 ζ(6), 0, 16 4ζ(3), 20ζ(5). 6ζ(2), 36ζ(4), 0, 50ζ(6), −1072ζ(8), −6912ζ(10). 6ζ(3), 18ζ(5), −18ζ(3)2 − 60ζ(6), 36ζ(3)3 + 360ζ(3)ζ(6) + 332ζ(9).

Remark 4.4. D(N, N − 1) is isomorphic to B(N − 1, N − 1), so in the case D(6, 5) we again have Apery’s recurrence for ζ(3) here. 5. Exceptional cases - E, F , G We provide computations of Apery constants only for a few of 23 exceptional homogeneous varieties, those with not too big spaces of cohomologies. X µ Apery numbers E(6, 6) 3 6ζ(4), 08 . E(6, 2) 6 03 , 18ζ(4), 90ζ(6), 07 , −3456ζ(10). E(7, 7) 3 −24ζ(5), 168ζ(9). E(8, 8) 11 120ζ(6), −1512ζ(10), . . . (of degrees 12, 16, 18, 22, 28). F (4, 1) 2 21ζ(4). F (4, 3) 8 −4ζ(2), 03 , −2ζ(4), −24ζ(5), −246ζ(6), 32ζ(2)ζ(5) + 60ζ(7), 2160ζ(2)ζ(7) − 144ζ(4)ζ(5). F (4, 4) 2 6ζ(4). Remark 5.1. There are two roots in the root system of G2 , taking factor by the parabolic subgroup associated with the smaller one we get a projective space, so later by G2 /P we denote the 5dimensional factor by another maximal parabolic subgroup. There is no literal Apery constants for G2 /P since this variety is minimal, so the only primitive cohomology class is 1, altough one may seek for almost solutions of quantum differential equation (strictly speaking Apery himself also considered such solutions). In [7] Golyshev considers this problem for Fano threefold V18 (i.e. a section of G2 /P by two hyperplanes) and using Beukers argument [3] and modularity of the quantum D-module for V18 shows that Apery number is equal to L√−3 (3) 6. Varieties with greater rank of Picard group, non-Calabi-Yau and Euler constant One may consider the same question for varieties X with higher Picard group. Canonicaly we should put H = −KX , but if we like, we could choose any H ∈ Pic(X). Even for such simple spaces as products of projective spaces one immidiately calculates some non-trivial Apery constants. X

µ Apery numbers 7

P2 × P2 3 P2 × P3 3

01 , 6ζ(2). 01 , 14 ζ(2). 3

In all these cases Apery numbers corresponding to all primitive divisors vanish. Van Straten’s calculation [14] relates monodromies of QDE for Fano fourfold X not to Chern numbers of the Fano, but to Chern numbers of it’s anticanonical Calabi-Yau hyperplane section Y . Probably Cfactors shouls correspond to c1 -factors in the Chern number, and since for Calaby-Yau c1 (Y ) = 0 we observe Euler constant is not involved. So one should consider something non-anticanonical. Let’s test the case H = O(1, 1) on P2 × P3 . Being exact, we restrict D-module to subtorus corresponding to H, and consider operator of quantum multiplication by H on it (subtorus associated with H is invariant with respect to vector field associated with H). (X, H) µ Apery numbers 2 2 3 (P × P , O(1, 1)) 3 −C, C +7ζ(2) . 2 7. Irrationality, special varieties and further speculations First of all let us note that both differential equations considered by Apery for the proofs of irrationality of ζ(2) and ζ(3) are essentially appeared in our computations as quantum differential equations of homogeneous varieties Gr(2, 5) and OGr(5, 10) = D(5, 4) (and isomorphic OGr(4, 9) = B(4, 4)). By essentially we mean the following proposition — Apery constants are invariant with respect to taking hyperplane section if the corresponding primitive classes survive: Proposition 7.1. Let X be a subcanonically embedded smooth Fano variety5 of index r > 1 i.e. X is embedded to the projective space by a linear system |H|, and −KX = rH. Consider a general hyperplane section Y — a subcanonically embedded smooth Fano variety of index r − 1. There is a restriction map γ → γ ∩ H from cohomologies of X to cohomologies of Y and by Hard Lefschetz theorem except possible of intermideate codimension all primitive classes of Y are restricted primitive classes of X. Consider a homogeneous primitive class of nonintermediate q codimension γ ∈ H (X). Then Apery numbers for γ calculated from QDE of X and Y coincide. Proof. By the quantum Lefschetz theorem of Givental-Kim-Gathmann we have a relation between q the I-series (solution of 1.1 associated with 1 ∈ H ) of X and Y : e.g. if r > 2 and Pic(X) = ZH Q and H2 (X, Z) = Zβ then d0 th coefficient of I − series of X should be multiplied by dHβ i=0 (H + i), if r 6 2 one should also do a change of coordinate. One may show the similar relation between solutions of 1.1 associated with γ and γ|Y : either directly repeating the arguments of original proof, or by Frobenius method of solving differential equation. So the limit of the ratio is the same.  One may rephrase the previous proposition in the following way Proposition 7.2. Apery class is functorial with respect to hyperplane sections. Proposition 7.2 is slightly stronger then 7.1: indeed, the intermediate primitive classes of X vanish restricted on Y , but also it states that ”parasitic” intermediate primitive classes of Y has Apery constant equal to 0. Following notations of [8] let’s call all smooth varieties related to each other by hyperplane section or deformation a strain, and if Y is a hyperplane section of X let’s call X an unsection of Y ; if Y has no unsections we call it a progenitor of the strain. The stability 5One

may state this proposition in higher generality, but we are going to use it for homogeneous spaces, and as stated it will be enough. 8

of Apery class is quite of the same nature as the stability of spectra in the strain described in [8]. Propositions 7.1 and 7.2 suggest to consider some kind of stable Apery class on the infinite hyperplane unsection. Such a stable framework of Gromov–Witten invariants was constructed by Przyalkowski for the case of quantum minimal Fano varieties in [13], using only KontsevichManin axioms. The next proposition shows that literaly this construction gives nothing from our perspective Proposition 7.3. If a Fano variety X is quantum minimal then all Apery constants vanish i.e. Apery class A is equal to 1. Proof. It is a trivial consequence of the definition of quantum minimality — since all primitive classes except 1 are quantum orthogonal to C[KX ] the operator of quantum multiplication by KX restricted to nonmaximal Lefschetz blocks coincides with the cup-product, in particular it is (k) nilpotent, so the associated solutions Aγ of QDE are polynomial in q i.e. their coefficients aγ vanish for k >> 0, hence the Apery number is 0.  Conjecture 7.4. The converse to 7.3 statement is true as well. So for our purposes the framework of [13] should be generalized taking into account the structure of Lefschetz decomposition. Another obstacle is geometrical nonliftability of varieties to higher dimensions — one can show both Grassmanian Gr(2, 5) (and any other Grassmanian except projective spaces and quadrics) and OGr(5, 10) are progenitors of their strains, i.e. cannot be represented as a hyperplane section of any nonsingular variety, this follows e.g. from the fact that these varieties are selfdual, but of course they are hyperplane sections of their cones. We insist that the quantum recursions for the progenitors Gr(2, 5) and OGr(5, 10) are the most natural in the strain, in particular in both cases we consider two exact solutions of the recursion, and in Apery’s case one considers an almost solution with polynomial error term — because for the linear sections of dimension 6 3 (6 5) the second Lefschetz block vanishes. One may ask a natural question whether any of the experimentaly or theoreticaly calculated Apery numbers (and their representations as the limits of the ratios of coefficients of two solutions of the recurrence) may be proven to be irrational by Apery’s argument. At least we know it works in two cases of Gr(2, 5) and OGr(5, 10). Remind that for irrationality of α = ζ(2) or α = ζ(3) one shows that (α − aqnγ ) is smaller then q1n , so we are interested in the sign of lim log(|α − aqnγ |) − log(qn ) (or equivalently in the sign of aγ (7.5) lim log log(|α − |) − log log qn . qn There were many attempts to find any other recurencies with this sign being negative, and most of them failed to the best of our knowledge. The quantum recursions we considered in this article is not an exception (we calculated convergence speed 7.5 numerically for n > 20). For example the convergence speed for ζ(2) approximation from Gr(2, N ) decreases as N grows, and is suitable only in the case of Gr(2, 5). So we come to the question: what is so special about Gr(2, 5) and OGr(5, 10)? One immideately reminds the famous theorem of Ein (see e.g. [15]) Theorem 7.6. Let X ⊂ PN be a smooth nondegenerate irreducible n-dimensional variety, such that X has the same dimension as it’s projectively dual X ∗ . Assume N > 3n . Then X is either a 2 hypersurface, or one of (1) a Segre variety P1 × Pr ⊂ P2r+1 9

(2) the Plucker embedding Gr(2, 5) ⊂ P9 (3) OGr(5, 10) Three last cases are selfdual: X ' X ∗ . Remark 7.7. For 7.6 we have the coincidence of the coherent and topological cohomologies (7.8)

q

N + 1 = dim H 0 (X, O(H)) = dim H (X)

In all 3 cases there are exactly two Lefschetz blocks, the codimensions of the grading of second Lefschetz block are corr. 1, 2 and 3. Remark 7.9. Apery number for P1 × Pr should approximate some multiple of C, but for r = 1, 2, 3 it is 0. As pointed out in section 6 we haven’t got any natural approximations for Euler constant in anticanonical Landau–Ginzburg model. From the other point of view, the variety P1 × Pr in the statement of the theorem 7.6 is not (sub)anticanonically embedded, but embedded by the linear system O(1, 1). Calculations of 6 are what we expect to be the quantum recursion for X embedded by O(1, 1), they indeed approximate C, but the speed of convergence is too slow. Either our guess is not correct (or not working here) or Landau–Ginzburg corresponding to the linear system |O(1, 1)| is something else. So the theorem 7.6 suggests the irrationality of Apery approximations are ruled by either selfduality or extremal defectiveness of the progenitor. Varieties 7.6 are related by the famous construction: let X be one of them, choose any point p ∈ X (they are homogeneous so all points are equivalent), then take an intersection of X with it’s tangent space Y = X ∩ Tp X. Then Y is a cone over the previous one: (7.10)

Tp Gr(2, 5) ∩ Gr(2, 5) = Cone(P1 × P2 )

(7.11)

Tp OGr(5, 10) ∩ OGr(5, 10) = Cone(Gr(2, 5))

In that way OGr(5, 10) can be ”lifted” one step further to Cartan variety E(6, 6) = E(6, 1): Tp E(6, 6) ∩ E(6, 6) = Cone(OGr(5, 10)). E(6, 6) is one of the four famous Severi varieties (or more general class of Scorza varieties) classified by Fyodor Zak in [15]: 3n+4

Theorem 7.12. Let X ⊂ PN = 2 be n-dimensional Severi variety i.e. X can be isomorphically projected to PN −1 . Then X is projectively equivalent to one of (1) the Veronese surface v2 (P2 ) ⊂ P5 (2) the Segre fourfold P2 × P2 ⊂ P8 (3) the Grassmanian Gr(2, 6) ⊂ P14 (4) the Cartan variety E(6, 6) ⊂ P26 Remark 7.13. Apart from the first case that should be correctly interpreted (e.g. taking symmetric square of D-module for P2 ), in the other 3 cases coincidence 7.8 holds (this is general fact for the closures of highest weight orbits of algebraic groups). The Lefschetz decompositions now consist of 3 blocks — first associated with 1, next one, and one block of length 1 in intermediate codimension. The last block has Apery number equal to 0. Neither of Severi varieties provides us with a fast enough approximation, but the speeds of convergence for them seem to be better then for arbitrary varieties. So it may be possible that these speeds are related with the defect of the variety (it is also supported by the fact that for Grassmanians defect decreases when N grows). 10

From the other perspective, when there are more then two Lefschetz blocks in the decomposition one may try to use the simultaneous Apery-type approximations of a tuple of zeta-polynomials as in the works of Zudilin. We would like to note that the recursion 1.1 contains more then one approximation of every Apery number appearing. Clearly speaking, in the definition of Apery numbers we considered the limit of the ratios of fundamental terms i.e. projections of two solutions A0 and Aγ to H 0 (X). It is natural to ask if we get anything from considering the limits of ratios of the other coordinates. Our experiments support the following (k)

(k)

Conjecture 7.14. Aγ is approximately equal to Apery(γ) · A0 as k → ∞. (k)

One may divide Aγ

(k)

by A0

q

in the nilpotent ring of H (X) and state the limit of such ratio

exists and is equal to Apery(γ) ∈ H 0 (X). For homogeneous γ2 the ratio

(k)

(Aγ ,γ2 ) (k)

(Aγ ,1)

grows as k codim γ2

and the cooordinates in the same Lefshetz block are linearly dependant. Finally let us provide some speculations explaining why the described behaviour is natural and also why zeta-values should appear. Assume for simplicity that the matrix of quantum multiplication by H has degree 1 in q (it is often the case for homogeneous varieties). Let M0 be the operator of cup-product by H and M1 be the degree 1 coefficient of quantum product by H. Then the quantum recursion is one-step: (7.15)

A(n) =

1 M0 M02 1 M1 An−1 = (1 + + 2 + . . . ) · M1 An−1 n − M0 n n n

Assume M0 and M1 commutes (actually, this is never true in our case). Then l

A

(l)

1Y M0 M02 = (1 + + 2 + . . . ) · M1l A(0) l! n=1 n n

Put l X X 1 M0k M0 M02 + 2 + . . . ) = exp( ). Nl = (1 + n n k nk n=1 n=1 k>1 l Y

Up to normalization lim Nl is Γ(1 + M0 ). Assume further that largest (by absolute value) eigenvalue α of M1 has the unique eigenvector β of multiplicity 1. Then A(l) is approximately equal to 1 C(A(0) , β) · · αl · Nl β l! Since M0 and M1 doesn’t commute there are additional terms from the commutators of Γ(1+M0 ) and M1 . Acknowledges. The database [6] of quantum cohomologies of homogeneous varieties was prepared jointly with V. Golyshev during our interest in the spectra of Fano varieties (inspired by [5]), and the interest to the Apery constant appeared after V. Golyshev’s deresonance computation [7] of the first Apery constant for Gr(2, N ). Author thanks Duco van Straten for the fruitful discussions and the invitation to the SFB/TR 45 program at Johannes Gutenberg Universitaet. 11

References [1] G. Almkvist, D. van Straten, W. Zudilin, Apery Limits of Differential Equations of Order 4 and 5 [2] R. Apery, Irrationalite de ζ(2) et ζ(3), Asterisque 61 (1979), 11–13 [3] F. Beukers, Irrationality proofs using modular forms., Journ´ees arithm´etiques, Besan¸con/France 1985, Ast´erisque 147/148, 271-283 (1987)., 1987. [4] W. Fulton, C. Woodward, On the quantum product of Schubert classes, J. Algebr. Geom. 13, No. 4 (2004), 641–661, arXiv:math.AG/0112183. [5] S. Galkin, V. Golyshev, Quantum cohomology of Grassmannians and cyclotomic fields, Russian Mathematical Surveys (2006), 61(1):171 [6] S. Galkin, V. Golyshev, Quantum cohomology of homogeneous varieties and noncyclotomic fields: PARI/GP database with quantum multiplications by ample Picard generator in homogeneous varieties of types A1 − A7, B2 − B7, C2 − C7, D4 − D7, E6, E7, F 4, G2., http://www.mi.ras.ru/∼galkin/work/u7.gp.bz2 [7] V. Golyshev, Rationality of Mukai varieties and Apery periods (in Russian), a draft [8] V. Golyshev, Spectra and strains, arXiv:0801.0432 [9] H.Iritani, Real and integral structures in quantum cohomology I: toric orbifolds, arXiv:0712.2204v2 [10] L.Katzarkov, M.Kontsevich, T.Pantev, Hodge theoretic aspects of mirror symmetry, arXiv:0806.0107 [11] LiE computer algebra system. http://www-math.univ-poitiers.fr/∼maavl/LiE [12] PARI/GP, version 2.3.3, Bordeaux, 2008, http://pari.math.u-bordeaux.fr/ [13] V. Przyjalkowski, Minimal Gromov–Witten ring, arXiv:0710.4084 [14] D. van Straten, a letter to Golyshev, 2007 [15] F. L. Zak, Severi varieties, Mathematics of the USSR-Sbornik (1986), 54(1):113,

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